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Given a random Bernoulli matrix $ A\in \{0,1\}^{m\times n} $, an integer $ 0<k < n $ and the vector $ y:=Ax $, where $ x \in \{0,1\}^n $ is of Hammingweight $ k $, the objective in the {\em Quantitative Group Testing} (QGT)problem is to recover $ x $. This problem is more difficult the smaller $m$ is.For parameter ranges of interest to us, known polynomial time algorithmsrequire values of $m$ that are much larger than $k$.  In this work, we define a seemingly easier problem that we refer to as {\emSubset Select}. Given the same input as in QGT, the objective in Subset Selectis to return a subset $ S \subseteq [n] $ of cardinality $ m $, such that forall $ i\in [n] $, if $ x_i = 1 $ then $ i\in S $. We show that if the squaresubmatrix of $A$ defined by the columns indexed by $S$ has nearly full rank,then from the solution of the Subset Select problem we can recover inpolynomial-time the solution $x$ to the QGT problem. We conjecture that forevery polynomial time Subset Select algorithm, the resulting output matrix willsatisfy the desired rank condition. We prove the conjecture for some classes ofalgorithms. Using this reduction, we provide some examples of how to improveknown QGT algorithms. Using theoretical analysis and simulations, wedemonstrate that the modified algorithms solve the QGT problem for values of $m $ that are smaller than those required for the original algorithms.

Quantitative Group Testing and the rank of random matrices